In this section, let

${\rho}_{1}>0$ be a constant. We denote by

$\mathcal{C}$ the set of all functions

$y:(-{\rho}_{1},{\rho}_{1})\to \mathbb{C}$ with the following properties:

- (a)
$y(x)$ is expressible by a power series ${\sum}_{m=0}^{\mathrm{\infty}}{b}_{m}{x}^{m}$ whose radius of convergence is at least ${\rho}_{1}$;

- (b)
There exists a constant

$K\ge 0$ such that

${\sum}_{m=0}^{\mathrm{\infty}}|{a}_{m}{x}^{m}|\le K|{\sum}_{m=0}^{\mathrm{\infty}}{a}_{m}{x}^{m}|$ for any

$x\in (-{\rho}_{1},{\rho}_{1})$, where

${a}_{m}=\sum _{k=0}^{m}[(k+2)(k+1){b}_{k+2}{p}_{m-k}+(k+1){b}_{k+1}{q}_{m-k}+{b}_{k}{r}_{m-k}]$

for all $m\in {\mathbb{N}}_{0}$ and ${p}_{0}\ne 0$.

**Lemma 3.1** *Given a sequence* $\{{a}_{m}\}$, *let* $\{{c}_{m}\}$ *be a sequence satisfying the recurrence formula* (3) *for all* $m\in {\mathbb{N}}_{0}$. *If* ${p}_{0}\ne 0$ *and* $n\ge 2$, *then* ${c}_{n}$ *is a linear combination of* ${a}_{0},{a}_{1},\dots ,{a}_{n-2}$, ${c}_{0}$, *and* ${c}_{1}$.

*Proof* We apply induction on

*n*. Since

${p}_{0}\ne 0$, if we set

$m=0$ in (3), then

${c}_{2}=\frac{1}{2{p}_{0}}{a}_{0}-\frac{{r}_{0}}{2{p}_{0}}{c}_{0}-\frac{{q}_{0}}{2{p}_{0}}{c}_{1},$

*i.e.*,

${c}_{2}$ is a linear combination of

${a}_{0}$,

${c}_{0}$, and

${c}_{1}$. Assume now that

*n* is an integer not less than 2 and

${c}_{i}$ is a linear combination of

${a}_{0},\dots ,{a}_{i-2}$,

${c}_{0}$,

${c}_{1}$ for all

$i\in \{2,3,\dots ,n\}$, namely,

${c}_{i}={\alpha}_{i}^{0}{a}_{0}+{\alpha}_{i}^{1}{a}_{1}+\cdots +{\alpha}_{i}^{i-2}{a}_{i-2}+{\beta}_{i}{c}_{0}+{\gamma}_{i}{c}_{1},$

where

${\alpha}_{i}^{0},\dots ,{\alpha}_{i}^{i-2}$,

${\beta}_{i}$,

${\gamma}_{i}$ are complex numbers. If we replace

*m* in (3) with

$n-1$, then

$\begin{array}{rcl}{a}_{n-1}& =& 2{c}_{2}{p}_{n-1}+{c}_{1}{q}_{n-1}+{c}_{0}{r}_{n-1}\\ +6{c}_{3}{p}_{n-2}+2{c}_{2}{q}_{n-2}+{c}_{1}{r}_{n-2}\\ +\cdots \\ +n(n-1){c}_{n}{p}_{1}+(n-1){c}_{n-1}{q}_{1}+{c}_{n-2}{r}_{1}\\ +(n+1)n{c}_{n+1}{p}_{0}+n{c}_{n}{q}_{0}+{c}_{n-1}{r}_{0}\\ =& (n+1)n{p}_{0}{c}_{n+1}+[n(n-1){p}_{1}+n{q}_{0}]{c}_{n}+\cdots \\ +(2{p}_{n-1}+2{q}_{n-2}+{r}_{n-3}){c}_{2}+({q}_{n-1}+{r}_{n-2}){c}_{1}+{r}_{n-1}{c}_{0},\end{array}$

which implies

$\begin{array}{rcl}{c}_{n+1}& =& \frac{1}{(n+1)n{p}_{0}}{a}_{n-1}-\frac{n(n-1){p}_{1}+n{q}_{0}}{(n+1)n{p}_{0}}{c}_{n}-\cdots \\ -\frac{2{p}_{n-1}+2{q}_{n-2}+{r}_{n-3}}{(n+1)n{p}_{0}}{c}_{2}-\frac{{q}_{n-1}+{r}_{n-2}}{(n+1)n{p}_{0}}{c}_{1}-\frac{{r}_{n-1}}{(n+1)n{p}_{0}}{c}_{0}\\ =& {\alpha}_{n+1}^{0}{a}_{0}+{\alpha}_{n+1}^{1}{a}_{1}+\cdots +{\alpha}_{n+1}^{n-1}{a}_{n-1}+{\beta}_{n+1}{c}_{0}+{\gamma}_{n+1}{c}_{1},\end{array}$

where ${\alpha}_{n+1}^{0},\dots ,{\alpha}_{n+1}^{n-1}$, ${\beta}_{n+1}$, ${\gamma}_{n+1}$ are complex numbers. That is, ${c}_{n+1}$ is a linear combination of ${a}_{0},{a}_{1},\dots ,{a}_{n-1}$, ${c}_{0}$, ${c}_{1}$, which ends the proof. □

In the following theorem, we investigate a kind of Hyers-Ulam stability of the linear differential equation (1). In other words, we answer the question whether there exists an exact solution near every approximate solution of (1). Since $x=0$ is an ordinary point of (1), we remark that ${p}_{0}\ne 0$.

**Theorem 3.2** *Let* $\{{c}_{m}\}$ *be a sequence of complex numbers satisfying the recurrence relation* (3)

*for all* $m\in {\mathbb{N}}_{0}$,

*where* (b)

*is referred for the value of* ${a}_{m}$,

*and let* ${\rho}_{2}$ *be the radius of convergence of the power series* ${\sum}_{m=0}^{\mathrm{\infty}}{c}_{m}{x}^{m}$.

*Define* ${\rho}_{3}=min\{{\rho}_{0},{\rho}_{1},{\rho}_{2}\}$,

*where* $(-{\rho}_{0},{\rho}_{0})$ *is the domain of the general solution to* (1).

*Assume that* $y:(-{\rho}_{1},{\rho}_{1})\to \mathbb{C}$ *is an arbitrary function belonging to* $\mathcal{C}$ *and satisfying the differential inequality* $|p(x){y}^{\u2033}(x)+q(x){y}^{\prime}(x)+r(x)y(x)|\le \epsilon $

(7)

*for all* $x\in (-{\rho}_{3},{\rho}_{3})$ *and for some* $\epsilon >0$.

*Let* ${\alpha}_{n}^{0},{\alpha}_{n}^{1},\dots ,{\alpha}_{n}^{n-2}$,

${\beta}_{n}$,

${\gamma}_{n}$ *be the complex numbers satisfying* ${c}_{n}={\alpha}_{n}^{0}{a}_{0}+{\alpha}_{n}^{1}{a}_{1}+\cdots +{\alpha}_{n}^{n-2}{a}_{n-2}+{\beta}_{n}{c}_{0}+{\gamma}_{n}{c}_{1}$

(8)

*for any integer* $n\ge 2$.

*If there exists a constant* $C>0$ *such that* $|{\alpha}_{n}^{0}{a}_{0}+{\alpha}_{n}^{1}{a}_{1}+\cdots +{\alpha}_{n}^{n-2}{a}_{n-2}|\le C|{a}_{n}|$

(9)

*for all integers* $n\ge 2$,

*then there exists a solution* ${y}_{h}:(-{\rho}_{3},{\rho}_{3})\to \mathbb{C}$ *of the linear homogeneous differential equation* (1)

*such that* $|y(x)-{y}_{h}(x)|\le CK\epsilon $

*for all* $x\in (-{\rho}_{3},{\rho}_{3})$, *where* *K* *is the constant determined in* (b).

*Proof* By the same argument presented in the proof of Theorem 2.1 with

${\sum}_{m=0}^{\mathrm{\infty}}{b}_{m}{x}^{m}$ instead of

${\sum}_{m=0}^{\mathrm{\infty}}{c}_{m}{x}^{m}$, we have

$p(x){y}^{\u2033}(x)+q(x){y}^{\prime}(x)+r(x)y(x)=\sum _{m=0}^{\mathrm{\infty}}{a}_{m}{x}^{m}$

(10)

for all

$x\in (-{\rho}_{3},{\rho}_{3})$. In view of (b), there exists a constant

$K\ge 0$ such that

$\sum _{m=0}^{\mathrm{\infty}}\left|{a}_{m}{x}^{m}\right|\le K\left|\sum _{m=0}^{\mathrm{\infty}}{a}_{m}{x}^{m}\right|$

(11)

for all $x\in (-{\rho}_{1},{\rho}_{1})$.

Moreover, by using (7), (10), and (11), we get

$\sum _{m=0}^{\mathrm{\infty}}\left|{a}_{m}{x}^{m}\right|\le K\left|\sum _{m=0}^{\mathrm{\infty}}{a}_{m}{x}^{m}\right|\le K\epsilon $

for any $x\in (-{\rho}_{3},{\rho}_{3})$. (That is, the radius of convergence of power series ${\sum}_{m=0}^{\mathrm{\infty}}{a}_{m}{x}^{m}$ is at least ${\rho}_{3}$.)

According to Theorem 2.1 and (10),

$y(x)$ can be written as

$y(x)={y}_{h}(x)+\sum _{n=0}^{\mathrm{\infty}}{c}_{n}{x}^{n}$

(12)

for all $x\in (-{\rho}_{3},{\rho}_{3})$, where ${y}_{h}(x)$ is a solution of the homogeneous differential equation (1). In view of Lemma 3.1, the ${c}_{n}$ can be expressed by a linear combination of the form (8) for each integer $n\ge 2$.

Since

${\sum}_{n=0}^{\mathrm{\infty}}{c}_{n}{x}^{n}$ is a particular solution of (2), if we set

${c}_{0}={c}_{1}=0$, then it follows from (8), (9), and (12) that

$|y(x)-{y}_{h}(x)|\le \sum _{n=0}^{\mathrm{\infty}}\left|{c}_{n}{x}^{n}\right|\le CK\epsilon $

for all $x\in (-{\rho}_{3},{\rho}_{3})$. □